Intermediate Algebra Practice Questions — ACT — Page 5

Question 81Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 = 10$ . The line $y = 3x$ intersects the circle at a point $(x, y)$ in the first quadrant. What is the value of $x + y$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3x

into the equation

x^2 + y^2 = 10

yields

x^2 + (3x)^2 = 10

. Simplifying this expression gives

x^2 + 9x^2 = 10

, which combines to

10x^2 = 10

. Dividing both sides by 10 results in

x^2 = 1

. Since the point of intersection lies in the first quadrant, we take the positive square root, so

x = 1

. Substituting

x = 1

back into

y = 3x

gives

y = 3(1) = 3

. The sum of the coordinates is

x + y = 1 + 3 = 4

.

Step-by-Step Solution

1

Substitute the linear expression for

y

into the equation of the circle.

x^2 + (3x)^2 = 10

Substituting one equation into another allows us to solve for a single variable.

2

Simplify the equation and solve for

x

.

10x^2 = 10 \Rightarrow x^2 = 1 \Rightarrow x = 1

Combining like terms gives

10x^2 = 10

, which simplifies to

x^2 = 1

. Since the intersection point is in the first quadrant,

x

must be positive.

3

Substitute the value of

x

back into the linear equation to find

y

.

y = 3(1) = 3

Using the value of

x

in the linear equation determines the corresponding

y

-coordinate.

4

Calculate the sum of the coordinates

x + y

.

1 + 3 = 4

The question asks for the value of the sum of the coordinates of the intersection point.

Key Concept

Solving a system of linear and non-linear (circular) equations using substitution.

Question 82Question →

A local bakery sells chocolate chip cookies for $1.50 each and oatmeal raisin cookies for$ 1.25 each. The table below shows the number of cookies sold on Monday and Tuesday:

Day	Chocolate Chip	Oatmeal Raisin
Monday	$80$	$60$
Tuesday	$95$	$70$

If the sales data is represented by the matrix $S = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix}$ and the prices are represented by the matrix $P = \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}$ , which of the following matrices represents the total revenue, in dollars, generated on Monday and Tuesday, respectively?

\begin{bmatrix} 190.00 \\ 223.75 \end{bmatrix}

\begin{bmatrix} 120.00 & 75.00 \\ 142.50 & 87.50 \end{bmatrix}

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

\begin{bmatrix} 238.75 \\ 177.50 \end{bmatrix}

\begin{bmatrix} 185.00 \\ 220.00 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}$

Answer

The matrix representing the revenue is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

The correct matrix is obtained by performing standard matrix multiplication of

S

(dimensions

2 \times 2

) and

P

(dimensions

2 \times 1

). The resulting matrix has dimensions

2 \times 1

. The first element is the sum of the products of the first row of

S

and the column of

P

:

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. The second element is the sum of the products of the second row of

S

and the column of

P

:

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

. This gives the matrix containing the values 195.00 and 230.00.

Step-by-Step Solution

1

Set up the matrix multiplication of sales and prices

Multiply the

2 \times 2

sales matrix

S

by the

2 \times 1

price matrix

P

:

S \times P = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix} \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}

Matrix multiplication aligns the quantities sold of each item with their respective unit prices to calculate the total revenue per day.

2

Perform the matrix multiplication for each row

Monday's revenue is

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. Tuesday's revenue is

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

.

Multiplying the elements of each row in the first matrix by the corresponding elements in the column of the second matrix, and then summing them, calculates the dot product for each day.

3

Write the results in a

2 \times 1

matrix

The final product matrix is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

.

The product of a

2 \times 2

matrix and a

2 \times 1

matrix is a

2 \times 1

matrix containing the total values for each row.

Key Concept

Matrix Multiplication in Word Problems

Question 83Question →

For the functions $f(x) = |x - 2| - 3$ and $g(x) = x^2 - 2x$ , what is the sum of all real values of $x$ for which $g(f(x)) = 0$ ?

2

4

8

12

-8

Show answer & explanation

Answer: 8

Answer

The sum of all real values of

x

is 8.

To solve

g(f(x)) = 0

, we substitute

f(x)

as the variable in

g(x)

, obtaining

(f(x))^2 - 2f(x) = 0

. Factoring this equation yields

f(x)(f(x) - 2) = 0

, giving two cases:

f(x) = 0

or

f(x) = 2

. For the first case,

|x - 2| - 3 = 0

simplifies to

|x - 2| = 3

, which has solutions

x = 5

and

x = -1

. For the second case,

|x - 2| - 3 = 2

simplifies to

|x - 2| = 5

, which has solutions

x = 7

and

x = -3

. Summing these four solutions gives

5 + (-1) + 7 + (-3) = 8

.

Step-by-Step Solution

1

Substitute

f(x)

into

g(x)

to set up the composite equation

g(f(x)) = 0

.

g(f(x)) = (f(x))^2 - 2f(x) = 0

To evaluate a function composition, the output of the inner function

f(x)

becomes the input of the outer function

g(x)

.

2

Factor the quadratic equation in terms of

f(x)

.

f(x)(f(x) - 2) = 0

, which means

f(x) = 0

or

f(x) = 2

.

Factoring allows us to split the quadratic equation into two separate linear cases.

3

Substitute

f(x) = |x - 2| - 3

into the first case,

f(x) = 0

, and solve for

x

.

|x - 2| - 3 = 0 \Rightarrow |x - 2| = 3 \Rightarrow x - 2 = 3

or

x - 2 = -3

, yielding

x = 5

and

x = -1

.

An absolute value equation

|u| = c

has two cases:

u = c

and

u = -c

when

c \geq 0

.

4

Substitute

f(x) = |x - 2| - 3

into the second case,

f(x) = 2

, and solve for

x

.

|x - 2| - 3 = 2 \Rightarrow |x - 2| = 5 \Rightarrow x - 2 = 5

or

x - 2 = -5

, yielding

x = 7

and

x = -3

.

We must solve the second case using both positive and negative scenarios for the absolute value.

5

Calculate the sum of all four real solutions.

5 + (-1) + 7 + (-3) = 8

The question asks for the sum of all real values of

x

that satisfy the composite equation.

Key Concept

Function Composition and Absolute Value Equations

Alternative Method

Instead of expanding the algebraic expressions, substitute a temporary variable

u = f(x)

into

g(u) = 0

to get

u^2 - 2u = 0

, which easily factors to

u = 0

or

u = 2

. Then, substitute

f(x)

back to solve

|x - 2| - 3 = 0

and

|x - 2| - 3 = 2

.

Estimated Time:1m 30s

Question 84Question →

In the quadratic equation $a x^2 + b x + c = 0$ , the coefficients $a$ , $b$ , and $c$ are real numbers, and $a = \frac{2}{3}$ . If one of the roots of this equation is $x = \frac{1}{2} - i\sqrt{2}$ , where $i = \sqrt{-1}$ , what is the value of the product $bc$ ?

-1

-\frac{7}{9}

-\frac{1}{2}

0

\frac{7}{9}

Show answer & explanation

Answer: $-1$

Answer

-1

For any quadratic equation with real coefficients, if a complex number is a root, its conjugate must also be a root. Thus, the two roots are

x_1 = \frac{1}{2} - i\sqrt{2}

and

x_2 = \frac{1}{2} + i\sqrt{2}

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 1 = -\frac{b}{a}

, which gives

b = -\frac{2}{3}

. The product of the roots is

x_1 x_2 = \frac{1}{4} - 2i^2 = \frac{9}{4} = \frac{c}{a}

, which gives

c = \frac{3}{2}

. The product

bc

is therefore

(-\frac{2}{3})(\frac{3}{2}) = -1

.

Step-by-Step Solution

1

Find the second root of the quadratic equation.

The second root is

x_2 = \frac{1}{2} + i\sqrt{2}

.

Since the quadratic equation has real coefficients, complex roots must occur in conjugate pairs.

2

Calculate the sum of the roots to find the coefficient

b

.

b = -\frac{2}{3}

The sum of the roots is

x_1 + x_2 = (\frac{1}{2} - i\sqrt{2}) + (\frac{1}{2} + i\sqrt{2}) = 1

. By Vieta's formulas,

x_1 + x_2 = -\frac{b}{a}

. Substituting

a = \frac{2}{3}

gives

1 = -\frac{b}{2/3} \implies b = -\frac{2}{3}

.

3

Calculate the product of the roots to find the coefficient

c

.

c = \frac{3}{2}

The product of the roots is

x_1 x_2 = (\frac{1}{2} - i\sqrt{2})(\frac{1}{2} + i\sqrt{2}) = (\frac{1}{2})^2 - (i\sqrt{2})^2 = \frac{1}{4} - 2i^2

. Since

i^2 = -1

, the product is \frac{1}{4} - 2(-1) = \frac{9}{4}

. By Vieta's formulas,

x_1 x_2 = \frac{c}{a}

. Substituting

a = \frac{2}{3}

gives \frac{9}{4} = \frac{c}{2/3} \implies c = \frac{3}{2}

.

4

Calculate the product

bc

.

bc = -1

Multiplying the values of

b

and

c

gives

bc = (-\frac{2}{3})(\frac{3}{2}) = -1

.

Key Concept

Relating the roots of a quadratic equation with real coefficients to its coefficients via Vieta's formulas and the Complex Conjugate Root Theorem.

Question 85Question →

A line and a parabola intersect at a point $(x, y)$ in the first quadrant of the standard $(x, y)$ coordinate plane. If the equation of the line is $y = x + 1$ and the equation of the parabola is $y = (x - 1)^2$ , what is the value of $x + y$ ?

3

5

1

7

9

Show answer & explanation

Answer: 7

Answer

7

To find the intersection point, we set the two equations equal to each other:

(x - 1)^2 = x + 1

. Expanding the left side gives

x^2 - 2x + 1 = x + 1

. Subtracting

x + 1

from both sides yields

x^2 - 3x = 0

. Factoring gives

x(x - 3) = 0

, so

x = 0

or

x = 3

. The first quadrant requires positive coordinates, so we choose

x = 3

. Substituting this back into either equation gives

y = 3 + 1 = 4

. Thus, the intersection point is

(3, 4)

, and the sum of the coordinates is

3 + 4 = 7

.

Step-by-Step Solution

1

Set the two equations equal to find the x-coordinates of the intersection points.

(x - 1)^2 = x + 1

Since both equations are equal to

y

, their right-hand sides must be equal at the points of intersection.

2

Expand the quadratic expression and simplify the equation.

x^2 - 2x + 1 = x + 1 \implies x^2 - 3x = 0

Expanding the binomial

(x-1)^2

yields

x^2 - 2x + 1

. Subtracting

x

and

1

from both sides simplifies the equation to standard quadratic form.

3

Factor the quadratic equation to solve for

x

.

x(x - 3) = 0 \implies x = 0

or

x = 3

Factoring out the greatest common factor

x

allows us to find the roots of the equation.

4

Determine which solution lies in the first quadrant and calculate the corresponding y-coordinate.

x = 3 \implies y = 3 + 1 = 4

The first quadrant requires both coordinates to be strictly positive. The solution

x = 0

gives

(0, 1)

, which is on the y-axis. Thus, we must use

x = 3

, which gives

(3, 4)

.

5

Calculate the sum

x + y

.

3 + 4 = 7

The question asks for the value of

x + y

for the first-quadrant intersection point.

Key Concept

Solving systems of linear and non-linear equations by substitution, factoring quadratic equations, and applying coordinate plane quadrant constraints.

Estimated Time:1m 0s

Question 86Question →

A geometric sequence has a first term of $\frac{1}{4}$ and a common ratio of $\frac{1}{2}$ . What is the 4th term of this sequence?

\frac{1}{32}

\frac{1}{64}

\frac{1}{512}

64

\frac{2}{3}

Show answer & explanation

Answer: $\frac{1}{32}$

Answer

\frac{1}{32}

The correct answer is found by substituting

a_1 = \frac{1}{4}

,

r = \frac{1}{2}

, and

n = 4

into the geometric sequence formula

a_n = a_1 \cdot r^{n-1}

. This gives

a_4 = \frac{1}{4} \cdot (\frac{1}{2})^3 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

.

Step-by-Step Solution

1

Identify the given components of the geometric sequence.

The first term is

a_1 = \frac{1}{4}

and the common ratio is

r = \frac{1}{2}

. We want to find the 4th term, so

n = 4

.

Before performing any calculations, we must identify the values of the variables needed for the geometric sequence formula.

2

Substitute the values into the general formula for the

n

-th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

.

a_4 = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^3

This formula relates the

n

-th term to the first term and the common ratio.

3

Evaluate the exponent first, and then multiply the fractions.

a_4 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

Following the order of operations, we raise the common ratio to the 3rd power before multiplying by the first term.

Key Concept

Finding the

n

-th term of a geometric sequence using the general formula

a_n = a_1 \cdot r^{n-1}

.

Question 87Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{(1 + 2i)^3}{2 - i}$ . What is the absolute value of $z$ ?

Show answer & explanation

Answer: 5

Answer

5

The complex number

z

simplifies to

-4 - 3i

. The absolute value of a complex number

a + bi

is defined as

\sqrt{a^2 + b^2}

, which gives

\sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5

. Alternatively, using properties of the modulus,

|z| = \frac{|1 + 2i|^3}{|2 - i|} = \frac{(\sqrt{1^2 + 2^2})^3}{\sqrt{2^2 + (-1)^2}} = \frac{(\sqrt{5})^3}{\sqrt{5}} = 5

.

Step-by-Step Solution

1

Expand the numerator

(1 + 2i)^3

-11 - 2i

To begin simplifying the complex fraction by expanding the cubed binomial in the numerator.

2

Divide the result by the denominator

2 - i

by multiplying both numerator and denominator by the complex conjugate

2 + i

-4 - 3i

To express the quotient in standard complex form

a + bi

.

3

Find the absolute value of the simplified complex number

z = -4 - 3i

5

To calculate the final magnitude of the complex number using

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Complex number operations, complex division, and absolute value (modulus) of complex numbers

Question 88Question →

A projectile is launched from ground level with an initial upward velocity of $v$ meters per second. The height of the projectile in meters after $t$ seconds is modeled by the equation $h(t) = -\frac{9}{2}t^2 + vt$ . A sensor is placed at a height of $32$ meters. What is the minimum integer value of $v$ for which the projectile is at or above the height of the sensor for a duration of at least $4$ seconds?

Show answer & explanation

Answer: 30

Answer

The minimum integer value of the initial upward velocity

v

is 30.

The height condition

h(t) \ge 32

leads to the quadratic inequality

9t^2 - 2vt + 64 \le 0

. The roots

t_1

and

t_2

of

9t^2 - 2vt + 64 = 0

represent the boundary times. The duration above

32

meters is the difference of the roots:

t_2 - t_1 = \frac{\sqrt{b^2 - 4ac}}{|a|} = \frac{\sqrt{4v^2 - 2304}}{9}

. Setting this to be at least

4

seconds gives

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

. Squaring both sides after multiplying by

9

yields

4v^2 - 2304 \ge 1296

, which simplifies to

4v^2 \ge 3600

, or

v^2 \ge 900

. Taking the positive square root gives

v \ge 30

, meaning the minimum integer value is

30

.

Step-by-Step Solution

1

Write the inequality representing the condition that the projectile is at or above the sensor height.

9t^2 - 2vt + 64 \le 0

The height function

h(t) = -\frac{9}{2}t^2 + vt

must be greater than or equal to

32

meters.

2

Use the formula for the difference of roots to represent the time duration.

t_2 - t_1 = \frac{\sqrt{4v^2 - 2304}}{9}

The duration of time the projectile is above the sensor is the difference between the two times at which the projectile crosses the sensor height.

3

Set the duration to be at least

4

seconds and solve for

v

.

v \ge 30

Solving the inequality

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

yields the range of initial velocities.

4

Find the smallest integer in the solution set.

30

Since the velocity must be an integer and at least

30

, the minimum possible integer value is

30

.

Key Concept

Solving quadratic inequalities and finding the difference of roots using the discriminant.

Question 89Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq 3$ , what is the real solution to the equation $\frac{x}{x-3} + \frac{2}{x} = \frac{3}{x-3}$ ?

Show answer & explanation

Answer: -2

Answer

The correct answer is -2.

Subtracting

\frac{x}{x-3}

from both sides yields

\frac{2}{x} = \frac{3-x}{x-3}

. Since

3-x = -(x-3)

, the right side simplifies to

-1

for all

x \neq 3

. The equation becomes

\frac{2}{x} = -1

, which gives

x = -2

. Since

-2

does not violate the domain constraints, it is the correct solution.

Step-by-Step Solution

1

Subtract

\frac{x}{x-3}

from both sides of the equation.

\frac{2}{x} = \frac{3-x}{x-3}

To group terms with common denominators on the same side.

2

Simplify the fraction on the right side of the equation.

\frac{2}{x} = -1

The numerator

3-x

is the negative of the denominator

x-3

, so their quotient is

-1

for all

x \neq 3

.

3

Solve the simplified equation

\frac{2}{x} = -1

for

x

.

x = -2

Multiplying both sides by

x

yields

2 = -x

, and dividing by

-1

gives

x = -2

.

4

Check the solution against the domain restrictions

x \neq 0

and

x \neq 3

.

The solution

x = -2

is valid.

Since

-2

is neither

0

nor

3

, it does not cause any denominator in the original equation to equal zero.

Key Concept

Solving rational equations by isolating terms with common denominators and checking for extraneous solutions.

Question 90Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = x^2 - 6x + 7$ and $g(x) = |2x - 5|$ . What is the sum of all real values of $x$ for which $f(g(x)) = 14$ ?

3

5

6

8

10

Show answer & explanation

Answer: 5

Answer

The sum of all possible real values of

x

is

5

.

The correct answer is

5

. Setting

f(g(x)) = 14

yields

(g(x))^2 - 6g(x) + 7 = 14

, which simplifies to

(g(x))^2 - 6g(x) - 7 = 0

. Factoring this quadratic gives

g(x) = 7

or

g(x) = -1

. Because

g(x) = |2x - 5|

cannot be negative, we discard the negative case. Solving

|2x - 5| = 7

gives

2x - 5 = 7 \Rightarrow x = 6

and

2x - 5 = -7 \Rightarrow x = -1

. The sum of these values is

6 + (-1) = 5

.

Step-by-Step Solution

1

Set up the composite equation

f(g(x)) = 14

by substituting

g(x)

into

f(x)

.

(g(x))^2 - 6g(x) + 7 = 14

This defines the relation between

g(x)

and the target output value.

2

Rearrange the equation into a standard quadratic form and solve for

g(x)

.

(g(x))^2 - 6g(x) - 7 = 0 \Rightarrow (g(x) - 7)(g(x) + 1) = 0

, so

g(x) = 7

or

g(x) = -1

.

Factoring the quadratic equation gives the possible values for the inner function

g(x)

.

3

Apply the definition of

g(x)

to solve for

x

and discard invalid cases.

Since

g(x) = |2x - 5|

must be non-negative, the case

g(x) = -1

has no real solutions. For the case

g(x) = 7

, we have

|2x - 5| = 7

.

The range of an absolute value function is

[0, \infty)

, making negative outputs impossible.

4

Solve the absolute value equation

|2x - 5| = 7

by setting up both positive and negative cases.

Either

2x - 5 = 7 \Rightarrow 2x = 12 \Rightarrow x = 6

, or

2x - 5 = -7 \Rightarrow 2x = -2 \Rightarrow x = -1

.

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

.

5

Sum the valid solutions for

x

.

6 + (-1) = 5

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Solving equations involving composite functions and absolute values

Estimated Time:2m 0s

Question 91Question →

A rectangular photograph has a length of $x$ inches and a width that is $1.5$ inches shorter than its length. If the area of the photograph is $10$ square inches, what is the length, in inches, of the photograph?

2.5

3.25

4

5

5.75

Show answer & explanation

Answer: 4

Answer

4

The correct length is 4. When the length is 4 inches, the width is 1.5 inches shorter, which is 2.5 inches. The area is the product of these dimensions: 4 * 2.5 = 10 square inches, which matches the given condition.

Step-by-Step Solution

1

Set up the equation for the area of the photograph.

x(x - 1.5) = 10

The area of a rectangle is found by multiplying its length (

x

) by its width (

x - 1.5

).

2

Distribute the

x

and subtract

10

from both sides to write the equation in standard form.

x^2 - 1.5x - 10 = 0

Standard form for a quadratic equation is

ax^2 + bx + c = 0

.

3

Multiply the entire equation by

2

to clear the decimal coefficient.

2x^2 - 3x - 20 = 0

Working with integer coefficients makes factoring or applying the quadratic formula easier.

4

Factor the quadratic expression.

(2x + 5)(x - 4) = 0

Factoring helps find the roots of the equation directly.

5

Solve for

x

by setting each factor to zero.

x = -2.5

or

x = 4

If the product of two numbers is zero, at least one of the numbers must be zero.

6

Exclude the negative solution.

x = 4

The length of a photograph must be a positive value.

Key Concept

Solving a quadratic equation by translating a word problem and factoring.

Question 92Question →

A sequence is defined by the formula $a_n = 2 + 3 \cdot \frac{2^n \cdot 2^{n-1}}{6}$ for all integers $n \geq 1$ . What is the value of the second term, $a_2$ ?

4

6

\frac{20}{3}

\frac{26}{7}

10

Show answer & explanation

Answer: $6$

Answer

The value of the second term is 6.

The correct answer is 6. When substituting

n = 2

into the formula, the numerator simplifies to

2^2 \cdot 2^1 = 8

. The fraction becomes

\frac{8}{6} = \frac{4}{3}

. Multiplying by 3 yields 4, and adding 2 yields 6.

Step-by-Step Solution

1

Substitute

n = 2

into the sequence formula.

a_2 = 2 + 3 \cdot \frac{2^2 \cdot 2^{2-1}}{6}

To find the second term, we replace

n

with 2 throughout the expression.

2

Simplify the exponents in the numerator.

2^2 \cdot 2^{2-1} = 4 \cdot 2^1 = 4 \cdot 2 = 8

Evaluate the exponential terms in the numerator before performing other operations.

3

Substitute the simplified numerator back and evaluate the fraction.

\frac{8}{6} = \frac{4}{3}

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

4

Multiply by 3 and add 2 to get the final answer.

2 + 3 \cdot \frac{4}{3} = 2 + 4 = 6

Perform multiplication before addition according to the order of operations.

Key Concept

Evaluating terms of a sequence given by an explicit formula involving exponent rules.

Estimated Time:1m 0s

Question 93Question →

A local bakery sells two types of muffins: blueberry and bran. Store 1 sells 30 blueberry muffins and 20 bran muffins. Store 2 sells 40 blueberry muffins and 35 bran muffins. Blueberry muffins cost $3 each, and bran muffins cost$ 2 each. This information is represented by the inventory matrix $M$ and the price matrix $P$ :

M = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix}

P = \begin{pmatrix} 3 \\ 2 \end{pmatrix}

Which of the following matrices represents the total revenue from muffin sales for Store 1 and Store 2, calculated using the matrix product $MP$ ?

\begin{pmatrix} 90 & 40 \\ 120 & 70 \end{pmatrix}

\begin{pmatrix} 120 \\ 185 \end{pmatrix}

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

\begin{pmatrix} 130 & 190 \end{pmatrix}

\begin{pmatrix} 150 \\ 210 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 130 \\ 190 \end{pmatrix}$

Answer

The column matrix with elements 130 and 190

The correct answer is the column matrix with elements 130 and 190. Multiplying the first row of the inventory matrix by the price column matrix yields the revenue for Store 1 (

30 \times 3 + 20 \times 2 = 130

). Multiplying the second row by the price column matrix yields the revenue for Store 2 (

40 \times 3 + 35 \times 2 = 190

). Since the product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 column matrix, the final result is correctly formatted.

Step-by-Step Solution

1

Set up the matrix multiplication of the 2 by 2 matrix and the 2 by 1 matrix.

MP = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix}

To calculate total revenue, we multiply the quantity matrix by the price matrix.

2

Calculate the first element of the resulting matrix (Store 1 total revenue) by taking the dot product of the first row of the inventory matrix and the price matrix.

30 \times 3 + 20 \times 2 = 90 + 40 = 130

This pairs the quantity of each muffin type sold at Store 1 with its respective price.

3

Calculate the second element of the resulting matrix (Store 2 total revenue) by taking the dot product of the second row of the inventory matrix and the price matrix.

40 \times 3 + 35 \times 2 = 120 + 70 = 190

This pairs the quantity of each muffin type sold at Store 2 with its respective price.

4

Assemble the resulting products into a 2 by 1 column matrix.

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

The product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 matrix.

Key Concept

Matrix multiplication of a 2 by 2 matrix and a 2 by 1 matrix in a real-world context.

Question 94Question →

For all real values of $t$ that satisfy the inequality $12 - 4|3 - 2t| \leq -8$ , which of the following inequalities represents the complete set of possible values of $t$ ?

t \leq -1

-1 \leq t \leq 4

t \geq -1

or

t \leq 4

t \leq -1

or

t \geq 4

t \leq 1

or

t \geq 2

Show answer & explanation

Answer: $t \leq -1$ or $t \geq 4$

Answer

t \leq -1

or

t \geq 4

The correct answer is

t \leq -1

or

t \geq 4

. To solve the inequality, we first subtract 12 from both sides to get

-4|3 - 2t| \leq -20

. Next, dividing both sides by

-4

and reversing the inequality sign gives

|3 - 2t| \geq 5

. This absolute value inequality splits into two cases:

3 - 2t \geq 5

or

3 - 2t \leq -5

. Solving the first case, we subtract 3 to get

-2t \geq 2

, and dividing by

-2

while reversing the inequality sign yields

t \leq -1

. Solving the second case, we subtract 3 to get

-2t \leq -8

, and dividing by

-2

while reversing the inequality sign yields

t \geq 4

. Combining these, we obtain the solution set

t \leq -1

or

t \geq 4

.

Step-by-Step Solution

1

Isolate the absolute value term by subtracting 12 from both sides of the inequality.

-4|3 - 2t| \leq -20

Before splitting an absolute value inequality, the absolute value expression must be isolated on one side.

2

Divide both sides of the inequality by

-4

and reverse the inequality sign.

|3 - 2t| \geq 5

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.

3

Split the absolute value inequality

|u| \geq c

(where

c > 0

) into two separate inequalities:

u \geq c

or

u \leq -c

.

3 - 2t \geq 5

or

3 - 2t \leq -5

An absolute value inequality of the form

|u| \geq c

represents values that are at least

c

units away from zero, which lie in two disjoint intervals.

4

Solve the first inequality:

3 - 2t \geq 5

.

-2t \geq 2 \implies t \leq -1

Subtracting 3 from both sides gives

-2t \geq 2

. Dividing by

-2

and reversing the inequality sign yields

t \leq -1

.

5

Solve the second inequality:

3 - 2t \leq -5

.

-2t \leq -8 \implies t \geq 4

Subtracting 3 from both sides gives

-2t \leq -8

. Dividing by

-2

and reversing the inequality sign yields

t \geq 4

.

6

Combine the solutions from both cases to express the complete solution set.

t \leq -1

or

t \geq 4

The complete solution set is the union of the solutions to both cases.

Key Concept

Solving absolute value inequalities of the form

|ax + b| \geq c

by translating them into compound inequalities and carefully reversing the inequality sign when multiplying or dividing by negative numbers.

Question 95Question →

The first term of an arithmetic sequence is $8$ , and the second term is $5$ . What is the $6$ th term of this sequence?

-15

-10

-7

-4

23

Show answer & explanation

Answer: -7

Answer

-7

The correct answer is

-7

. Since the sequence is arithmetic, it changes by a constant common difference,

d

, with each step. Calculating

d

gives

5 - 8 = -3

. To find the

6

th term, we start at the first term,

8

, and add the common difference

5

times (representing the steps from the first to the sixth term):

8 + 5(-3) = 8 - 15 = -7

.

Step-by-Step Solution

1

Determine the common difference,

d

, of the arithmetic sequence.

d = -3

Subtract the first term from the second term:

5 - 8 = -3

.

2

Set up the equation for the

n

th term of an arithmetic sequence,

a_n = a_1 + (n-1)d

.

a_6 = 8 + (6-1)(-3)

We want to find the 6th term (

n = 6

) starting with

a_1 = 8

and

d = -3

.

3

Perform the operations to find the final value.

a_6 = -7

First multiply

5 \times (-3) = -15

, then add to

8

to get

8 - 15 = -7

.

Key Concept

Finding a specific term of an arithmetic sequence using its first term and common difference.

Estimated Time:45s

Question 96Question →

What is the real solution to the equation $\sqrt{3x + 1} = x - 3$ ?

1

1 and 8

8

-2 and 5

-2

Show answer & explanation

Answer: 8

Answer

The only real solution is 8.

The correct answer is the single value 8. Squaring both sides of the equation

\sqrt{3x+1} = x-3

leads to the quadratic equation

x^2-9x+8=0

, which factors as

(x-8)(x-1)=0

. This yields potential solutions of 8 and 1. Checking 8 in the original equation gives

\sqrt{25} = 5

, which is correct. Checking 1 in the original equation gives

\sqrt{4} = -2

, which is incorrect because the principal square root must be non-negative. Therefore, 1 is an extraneous solution, and 8 is the only valid solution.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

3x + 1 = (x - 3)^2

To clear the square root and obtain a polynomial equation.

2

Expand the right side and move all terms to the right side to set the quadratic equation to zero.

x^2 - 9x + 8 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting

3x

and

1

from both sides gives the standard form of the quadratic equation.

3

Factor the quadratic equation.

(x - 8)(x - 1) = 0

Finding two numbers that multiply to 8 and add to -9, which are -8 and -1, allows us to find potential solutions

x = 8

and

x = 1

.

4

Check both potential solutions in the original equation to identify extraneous solutions.

Checking

x = 8

:

\sqrt{3(8) + 1} = \sqrt{25} = 5

and

8 - 3 = 5

(True). Checking

x = 1

:

\sqrt{3(1) + 1} = \sqrt{4} = 2

and

1 - 3 = -2

(False).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Key Concept

Solving radical equations and identifying extraneous solutions

Estimated Time:1m 30s

Question 97Question →

The first three terms of a geometric sequence of positive numbers are $x$ , $y$ , and $z$ , in that order. An arithmetic sequence has first three terms $x$ , $y$ , and $z - 4$ , in that order. If $x = 4$ , what is the value of $y$ ?

2

4

6

8

16

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By defining the geometric sequence terms as

4, y, z

, we have the property

y^2 = 4z

. For the arithmetic sequence

4, y, z-4

, the common difference property gives

y - 4 = z - 4 - y

, which simplifies to

2y = z

. Substituting

z = 2y

into the first equation yields

y^2 = 8y

. Since all terms must be positive, dividing by

y

gives

y = 8

.

Step-by-Step Solution

1

Set up the equation for the geometric sequence.

y^2 = 4z

Since

4, y, z

is a geometric sequence, the ratio of consecutive terms must be equal:

\frac{y}{4} = \frac{z}{y}

, which simplifies to

y^2 = 4z

.

2

Set up the equation for the arithmetic sequence.

2y = z

Since

4, y, z - 4

is an arithmetic sequence, the difference between consecutive terms must be equal:

y - 4 = (z - 4) - y

. Simplifying this gives

2y = z

.

3

Substitute the arithmetic equation into the geometric equation.

y^2 = 8y

Substituting

z = 2y

into

y^2 = 4z

yields

y^2 = 4(2y) = 8y

.

4

Solve for the variable

y

.

y = 8

Rearranging the equation gives

y^2 - 8y = 0

, which factors as

y(y - 8) = 0

. Since the sequence consists of positive numbers,

y

must be positive, so

y = 8

.

Key Concept

Relating arithmetic and geometric sequence properties to solve a system of non-linear equations

Question 98Question →

If $i$ represents the imaginary unit, and the complex number $z$ satisfies the equation $z(2 + i) - 5i^{97} = \frac{25}{3 - 4i}$ , what is the value of $z^2$ ?

-18i

-2i

16

18i

50i

Show answer & explanation

Answer: 18i

Answer

18i

Simplifying the original equation yields the complex number

z = 3 + 3i

. Squaring this number gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

.

Step-by-Step Solution

1

Simplify the fraction on the right side of the equation by multiplying the numerator and denominator by the complex conjugate of the denominator,

3 + 4i

.

The fraction simplifies to

3 + 4i

.

Multiplying by the conjugate rationalizes the denominator:

\frac{25}{3 - 4i} = \frac{25(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{25(3 + 4i)}{9 + 16} = 3 + 4i

.

2

Simplify the power of the imaginary unit,

i^{97}

, by dividing the exponent by 4 to find the remainder.

5i^{97} = 5i

Since

97 = 4 \times 24 + 1

, the expression simplifies as

i^{97} = (i^4)^{24} \cdot i = 1^{24} \cdot i = i

.

3

Substitute the simplified expressions back into the original equation and isolate the term containing

z

.

z(2 + i) = 3 + 9i

Substituting gives

z(2 + i) - 5i = 3 + 4i

. Adding

5i

to both sides yields

z(2 + i) = 3 + 9i

.

4

Solve for

z

by dividing both sides by

2 + i

, then simplify by multiplying by the conjugate of the denominator,

2 - i

.

z = 3 + 3i

Performing the division:

z = \frac{3 + 9i}{2 + i} = \frac{(3 + 9i)(2 - i)}{(2 + i)(2 - i)} = \frac{6 - 3i + 18i - 9i^2}{4 - i^2} = \frac{15 + 15i}{5} = 3 + 3i

.

5

Calculate the value of

z^2

by squaring the complex number

3 + 3i

.

z^2 = 18i

Squaring the binomial gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

.

Key Concept

Solving equations containing complex numbers by performing operations such as multiplication, division using complex conjugates, and simplifying powers of

i

.

Estimated Time:2m 30s

Question 99Question →

For the functions $f(x) = \frac{18}{x - 2}$ and $g(x) = x^2 - 1$ , what is the value of $f(g(3))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the composite function

f(g(3))

is 3.

To evaluate the composite function

f(g(3))

, we start by evaluating the inner function

g(3) = 3^2 - 1 = 8

. We then substitute this result into the outer function, giving

f(8) = \frac{18}{8 - 2} = \frac{18}{6} = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(3)

8

For any composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input value.

2

Evaluate the outer function

f(8)

3

Substitute the output value of the inner function as the input for the outer function

f(x)

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 100Question →

Consider the system of equations below:

\begin{aligned} x^2 + y^2 &= 9 \\ y &= x - 3 \end{aligned}

If $(x, y)$ is a solution to the system such that $x > 0$ , what is the value of $x + y$ ?

3

-3

0

6

9

Show answer & explanation

Answer: 3

Answer

3

The correct answer is

3

. Substituting

y = x - 3

into

x^2 + y^2 = 9

gives

x^2 + (x - 3)^2 = 9

. Expanding this yields

2x^2 - 6x = 0

, which factors as

2x(x - 3) = 0

. Since

x > 0

, the

x

-coordinate must be

3

. Substituting

x = 3

back into the linear equation gives

y = 3 - 3 = 0

. Therefore,

x + y = 3 + 0 = 3

.

Step-by-Step Solution

1

Substitute the linear equation into the quadratic equation.

x^2 + (x - 3)^2 = 9

This reduces the system of two variables to a single quadratic equation in terms of

x

.

2

Expand the squared binomial and simplify the equation.

x^2 + (x^2 - 6x + 9) = 9 \implies 2x^2 - 6x = 0

Expanding

(x - 3)^2

yields

x^2 - 6x + 9

. Subtracting

9

from both sides simplifies the equation.

3

Factor the quadratic equation to solve for

x

.

2x(x - 3) = 0 \implies x = 0

or

x = 3

Factoring out the greatest common factor

2x

reveals the roots of the quadratic equation.

4

Apply the constraint

x > 0

to find the correct

x

-coordinate.

x = 3

The problem specifies that the solution

(x, y)

must have

x > 0

.

5

Find the corresponding

y

-coordinate and calculate

x + y

.

y = 3 - 3 = 0 \implies x + y = 3 + 0 = 3

Substituting

x = 3

into the linear equation gives

y = 0

, and summing them gives the required value.

Key Concept

Solving systems of linear and quadratic equations by substitution

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